a^2-a-14=3a-10

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Solution for a^2-a-14=3a-10 equation:


Simplifying
a2 + -1a + -14 = 3a + -10

Reorder the terms:
-14 + -1a + a2 = 3a + -10

Reorder the terms:
-14 + -1a + a2 = -10 + 3a

Solving
-14 + -1a + a2 = -10 + 3a

Solving for variable 'a'.

Reorder the terms:
-14 + 10 + -1a + -3a + a2 = -10 + 3a + 10 + -3a

Combine like terms: -14 + 10 = -4
-4 + -1a + -3a + a2 = -10 + 3a + 10 + -3a

Combine like terms: -1a + -3a = -4a
-4 + -4a + a2 = -10 + 3a + 10 + -3a

Reorder the terms:
-4 + -4a + a2 = -10 + 10 + 3a + -3a

Combine like terms: -10 + 10 = 0
-4 + -4a + a2 = 0 + 3a + -3a
-4 + -4a + a2 = 3a + -3a

Combine like terms: 3a + -3a = 0
-4 + -4a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -4a + 4 + a2 = 0 + 4

Reorder the terms:
-4 + 4 + -4a + a2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -4a + a2 = 0 + 4
-4a + a2 = 0 + 4

Combine like terms: 0 + 4 = 4
-4a + a2 = 4

The a term is -4a.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4a + 4 + a2 = 4 + 4

Reorder the terms:
4 + -4a + a2 = 4 + 4

Combine like terms: 4 + 4 = 8
4 + -4a + a2 = 8

Factor a perfect square on the left side:
(a + -2)(a + -2) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(a + -2) equal to 2.828427125 and -2.828427125.

Subproblem 1

a + -2 = 2.828427125 Simplifying a + -2 = 2.828427125 Reorder the terms: -2 + a = 2.828427125 Solving -2 + a = 2.828427125 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 2.828427125 + 2 Combine like terms: -2 + 2 = 0 0 + a = 2.828427125 + 2 a = 2.828427125 + 2 Combine like terms: 2.828427125 + 2 = 4.828427125 a = 4.828427125 Simplifying a = 4.828427125

Subproblem 2

a + -2 = -2.828427125 Simplifying a + -2 = -2.828427125 Reorder the terms: -2 + a = -2.828427125 Solving -2 + a = -2.828427125 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -2.828427125 + 2 Combine like terms: -2 + 2 = 0 0 + a = -2.828427125 + 2 a = -2.828427125 + 2 Combine like terms: -2.828427125 + 2 = -0.828427125 a = -0.828427125 Simplifying a = -0.828427125

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.828427125, -0.828427125}

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